Primal-Dual Interior-Point Algorithms with Dynamic Step-Size Based on Kernel Functions for Linear Programming

In this paper, primal-dual interior-point algorithm with dynamic step size is implemented for linear programming （LP） problems. The algorithms are based on a few kernel functions, including both serf-regular functions and non-serf-regular ones. The dynamic step size is compared with fixed step size for the algorithms in inner iteration of Newton step. Numerical tests show that the algorithms with dynaraic step size are more efficient than those with fixed step size.

A Primal-Dual Infeasible-Interior-Point Algorithm for Multiple Objective Linear Programming Problems

A primal-dual infeasible interior point algorithm for multiple objective linear programming (MOLP) problems was presented. In contrast to the current MOLP algorithm. moving through the interior of polytope but not confining the iterates within the feasible region in our proposed algorithm result in a solution approach that is quite different and less sensitive to problem size, so providing the potential to dramatically improve the practical computation effectiveness.

A Primal-dual Interior Point Method for Nonlinear Programming

In this paper, we propose a primal-dual interior point method for solving general constrained nonlinear programming problems. To avoid the situation that the algorithm we use may converge to a saddle point or a local maximum, we utilize a merit function to guide the iterates toward a local minimum. Especially, we add the parameter ε to the Newton system when calculating the decrease directions. The global convergence is achieved by the decrease of a merit function. Furthermore, the numerical results confirm that the algorithm can solve this kind of problems in an efficient way.

A predictor-corrector interior-point algorithmfor monotone variational inequality problems

Mehrotra's recent suggestion of a predictor corrector variant of primal dual interior point method for linear programming is currently the interior point method of choice for linear programming. In this work the authors give a predictor corrector interior point algorithm for monotone variational inequality problems. The algorithm was proved to be equivalent to a level 1 perturbed composite Newton method. Computations in the algorithm do not require the initial iteration to be feasible. Numerical results of experiments are presented.

AN INFEASIBLE-INTERIOR-POINT PREDICTOR-CORRECTOR ALGORITHM FOR THE SECOND-ORDER CONE PROGRAM

A globally convergent infeasible-interior-point predictor-corrector algorithm is presented for the second-order cone programming （SOCP） by using the Alizadeh- Haeberly-Overton （AHO） search direction. This algorithm does not require the feasibility of the initial points and iteration points. Under suitable assumptions, it is shown that the algorithm can find an -approximate solution of an SOCP in at most O（√n ln（ε0/ε）） iterations. The iteration-complexity bound of our algorithm is almost the same as the best known bound of feasible interior point algorithms for the SOCP.

A POLYNOMIAL PREDICTOR-CORRECTOR INTERIOR-POINT ALGORITHM FOR CONVEX QUADRATIC PROGRAMMING

This article presents a polynomial predictor-corrector interior-point algorithm for convex quadratic programming based on a modified predictor-corrector interior-point algorithm. In this algorithm, there is only one corrector step after each predictor step, where Step 2 is a predictor step and Step 4 is a corrector step in the algorithm. In the algorithm, the predictor step decreases the dual gap as much as possible in a wider neighborhood of the central path and the corrector step draws iteration points back to a narrower neighborhood and make a reduction for the dual gap. It is shown that the algorithm has O（√nL） iteration complexity which is the best result for convex quadratic programming so far.

An Improved Affine-Scaling Interior Point Algorithm for Linear Programming

In this paper, an Improved Affine-Scaling Interior Point Algorithm for Linear Programming has been proposed. Computational results of selected practical problems affirming the proposed algorithm have been provided. The proposed algorithm is accurate, faster and therefore reduces the number of iterations required to obtain an optimal solution of a given Linear Programming problem as compared to the already existing Affine-Scaling Interior Point Algorithm. The algorithm can be very useful for development of faster software packages for solving linear programming problems using the interior-point methods.

Complexity analysis of interior-point algorithm based on a new kernel function for semidefinite optimization

Interior-point methods （IPMs） for linear optimization （LO） and semidefinite optimization （SDO） have become a hot area in mathematical programming in the last decades. In this paper, a new kernel function with simple algebraic expression is proposed. Based on this kernel function, a primal-dual interior-point methods （IPMs） for semidefinite optimization （SDO） is designed. And the iteration complexity of the algorithm as O（n^3/4 log n/ε） with large-updates is established. The resulting bound is better than the classical kernel function, with its iteration complexity O（n log n/ε） in large-updates case.

Power System Reactive Power Optimization Based on Fuzzy Formulation and Interior Point Filter Algorithm

Considering the soft constraint characteristics of voltage constraints, the Interior-Point Filter Algorithm is applied to solve the formulation of fuzzy model for the power system reactive power optimization with a large number of equality and inequality constraints. Based on the primal-dual interior-point algorithm, the algorithm maintains an updating “filter” at each iteration in order to decide whether to admit correction of iteration point which can avoid effectively oscillation due to the conflict between the decrease of objective function and the satisfaction of constraints and ensure the global convergence. Moreover, the “filter” improves computational efficiency because it filters the unnecessary iteration points. The calculation results of a practical power system indicate that the algorithm can effectively deal with the large number of inequality constraints of the fuzzy model of reactive power optimization and satisfy the requirement of online calculation which realizes to decrease the network loss and maintain specified margins of voltage.

Predictor-corrector interior-point algorithm for linearly constrained convex programming

Active set method and gradient projection method are curre nt ly the main approaches for linearly constrained convex programming. Interior-po int method is one of the most effective choices for linear programming. In the p aper a predictor-corrector interior-point algorithm for linearly constrained c onvex programming under the predictor-corrector motivation was proposed. In eac h iteration, the algorithm first performs a predictor-step to reduce the dualit y gap and then a corrector-step to keep the points close to the central traject ory. Computations in the algorithm only require that the initial iterate be nonn egative while feasibility or strict feasibility is not required. It is proved th at the algorithm is equivalent to a level-1 perturbed composite Newton method. Numerical experiments on twenty-six standard test problems are made. The result s show that the proposed algorithm is stable and robust.

Optimal Adjustment Algorithm for <i>p</i>Coordinates and The Starting Point in Interior Point Methods

Optimal adjustment algorithm for p coordinates is a generalization of the optimal pair adjustment algorithm for linear programming, which in turn is based on von Neumann’s algorithm. Its main advantages are simplicity and quick progress in the early iterations. In this work, to accelerate the convergence of the interior point method, few iterations of this generalized algorithm are applied to the Mehrotra’s heuristic, which determines the starting point for the interior point method in the PCx software. Computational experiments in a set of linear programming problems have shown that this approach reduces the total number of iterations and the running time for many of them, including large-scale ones.

A PREDICTOR-CORRECTOR INTERIOR-POINT ALGORITHM FOR CONVEX QUADRATIC PROGRAMMING

The simplified Newton method, at the expense of fast convergence, reduces the work required by Newton method by reusing the initial Jacobian matrix. The composite Newton method attempts to balance the trade-off between expense and fast convergence by composing one Newton step with one simplified Newton step. Recently, Mehrotra suggested a predictor-corrector variant of primal-dual interior point method for linear programming. It is currently the interiorpoint method of the choice for linear programming. In this work we propose a predictor-corrector interior-point algorithm for convex quadratic programming. It is proved that the algorithm is equivalent to a level-1 perturbed composite Newton method. Computations in the algorithm do not require that the initial primal and dual points be feasible. Numerical experiments are made.

A POSITIVE INTERIOR-POINT ALGORITHM FOR NONLINEAR COMPLEMENTARITY PROBLEMS

A new iterative method,which is called positive interior-point algorithm,is presented for solving the nonlinear complementarity problems.This method is of the desirable feature of robustness.And the convergence theorems of the algorithm is established.In addition,some numerical results are reported.

单调线性权互补问题的新全牛顿步可行内点算法

提出求解单调线性权互补问题(WLCP)的全牛顿步可行内点算法。基于线性优化的连续可微函数,给出中心方程的新等价形式,接着运用牛顿法求解定义中心路径的等价方程组,从而得到单调WLCP的新搜索方向。沿该搜索方向使用全牛顿步,无需进行线搜索。通过适当选取参数,分析了全牛顿步的严格可行性,证得算法是二次收敛的且具有多项式时间迭代复杂度。最后数值实验结果表明算法有效。

Novel Kernel Function With a Hyperbolic Barrier Term to Primal-dual Interior Point Algorithm for SDP Problems

In this paper,we introduce for the first time a new eligible kernel function with a hyperbolic barrier term for semidefinite programming(SDP).This add a new type of functions to the class of eligible kernel functions.We prove that the interior-point algorithm based on the new kernel function meets O(n3/4 logε/n)iterations as the worst case complexity bound for the large-update method.This coincides with the complexity bound obtained by the first kernel function with a trigonometric barrier term proposed by El Ghami et al.in2012,and improves with a factor n(1/4)the obtained iteration bound based on the classic kernel function.We present some numerical simulations which show the effectiveness of the algorithm developed in this paper.

A Class of New Large-Update Primal-Dual Interior-Point Algorithms for P＊（k） Nonlinear Complementarity Problems

In this paper we propose a class of new large-update primal-dual interior-point algorithms for P.（k） nonlinear complementarity problem （NCP）, which are based on a class of kernel functions investigated by Bai et al. in their recent work for linear optimization （LO）. The arguments for the algorithms are followed as Peng et al.＇s for P.（n） complementarity problem based on the self-regular functions [Peng, J., Roos, C., Terlaky, T.： Self-Regularity： A New Paradigm for Primal-Dual Interior- Point Algorithms, Princeton University Press, Princeton, 2002]. It is worth mentioning that since this class of kernel functions includes a class of non-self-regular functions as special case, so our algorithms are different from Peng et al.＇s and the corresponding analysis is simpler than theirs. The ultimate goal of the paper is to show that the algorithms based on these functions have favorable polynomial complexity.

A PRIMAL-DUAL FIXED POINT ALGORITHM FOR MULTI-BLOCK CONVEX MINIMIZATION

We have proposed a primal-dual fixed point algorithm （PDFP） for solving minimiza- tion of the sum of three convex separable functions, which involves a smooth function with Lipschitz continuous gradient, a linear composite nonsmooth function, and a nonsmooth function. Compared with similar works, the parameters in PDFP are easier to choose and are allowed in a relatively larger range. We will extend PDFP to solve two kinds of separable multi-block minimization problems, arising in signal processing and imaging science. This work shows the flexibility of applying PDFP algorithm to multi-block prob- lems and illustrates how practical and fully splitting schemes can be derived, especially for parallel implementation of large scale problems. The connections and comparisons to the alternating direction method of multiplier （ADMM） are also present. We demonstrate how different algorithms can be obtained by splitting the problems in different ways through the classic example of sparsity regularized least square model with constraint. In particular, for a class of linearly constrained problems, which are of great interest in the context of multi-block ADMM, can be also solved by PDFP with a guarantee of convergence. Finally, some experiments are provided to illustrate the performance of several schemes derived by the PDFP algorithm.

Two new predictor-corrector algorithms for second-order cone programming

Based on the ideas of infeasible interior-point methods and predictor-corrector algorithms, two interior-point predictor-corrector algorithms for the second-order cone programming （SOCP） are presented. The two algorithms use the Newton direction and the Euler direction as the predictor directions, respectively. The corrector directions belong to the category of the Alizadeh-Haeberly-Overton （AHO） directions. These algorithms are suitable to the cases of feasible and infeasible interior iterative points. A simpler neighborhood of the central path for the SOCP is proposed, which is the pivotal difference from other interior-point predictor-corrector algorithms. Under some assumptions, the algorithms possess the global, linear, and quadratic convergence. The complexity bound O（rln（εo/ε）） is obtained, where r denotes the number of the second-order cones in the SOCP problem. The numerical results show that the proposed algorithms are effective.

P∗(κ)-线性权互补问题的一种全牛顿步可行内点算法

本文提出一种求解P∗(κ)-线性权互补问题(LWCP)的新全牛顿步可行内点算法.首先基于一个连续可微的核函数,构造新代数等价变换,得到光滑中心路径的等价形式.然后沿着搜索方向使用全牛顿步,无需进行线搜索,节省运行内存.最后分析算法的可行性及收敛性,并通过数值算例验证算法的有效性.

非线性方程组的仿射尺度内点信赖域算法

很多领域研究寻优问题时,所采用的寻优算法普遍存在全局搜索能力差、收敛速度慢的问题,导致求出的解无法达到最优。针对上述问题,研究了一种非线性方程组的仿射尺度内点信赖域算法。构建目标最小化或者目标最大化非线性方程组,并针对方程组设置等式或者不等式约束条件;在约束条件下,利用仿射尺度内点信赖域算法求取非线性方程组最优解;将所研究算法应用到有功优化当中,以线损最小化和电压偏差最小化构建非线性方程组,并为其设置四个约束条件,利用仿射尺度内点信赖域算法求取最优解。实验结果表明:与自适应粒子群算法、樽海鞘群算法以及改进差分灰狼算法相比,所研究算法应用下,线损以及电压偏差均要更小,说明仿射尺度内点信赖域算法的求解结果更优,算法的寻优能力更强。